Approximaths

Approximaths

  • Perturbation Theory – Case Study 2

    We would like to solve a very simple problem. This time we will set a power of \epsilon:

    \displaystyle x - 2 = 0 \quad \text{Problem}
    \displaystyle x - 2\epsilon^2 = 0 \quad \text{Insert }\epsilon^2
    \displaystyle x = 0 \quad \text{Set }\epsilon^2 = 0
    \displaystyle (a_0 + a_1 \epsilon + a_2 \epsilon^2 + \dots) - 2\epsilon^2 = 0 \quad \text{Insert }x_{\epsilon}
    \displaystyle a_0 + a_1 \epsilon + (a_2 - 2)\epsilon^2 + \dots = 0 + 0 + 0 + \dots

    By identifying coefficients of equal powers of \epsilon (uniqueness of power series expansion), we obtain:

    \displaystyle \implies a_0 = 0,\ a_1 = 0,\ a_2 = 2,\ \dots
    \displaystyle x_{\epsilon} = 0 + 0 + 2\epsilon^2 + \dots
    \displaystyle x_{\epsilon} = 2\epsilon^2

    Evaluating the solution at \epsilon = 1 recovers the original problem and gives x = 2. In the figure below we visualize the solution of x - 2\epsilon^2 = 0 as a function of \epsilon.

    For this very simple example, we essentially chose a placement of \epsilon such that the unperturbed problem (for \epsilon = 0) becomes trivial. We also have many choices for the power of \epsilon. In this example, the perturbation series terminates and yields the exact solution.

  • Perturbation Theory – Case Study 1

    Since then, we have presented some methods to sum series (Padé approximants, Euler summation, Borel summation, Borel-Écalle summation, generic summation and Zeta summation). We have also seen techniques to accelerate convergence (Shanks transformation and Richardson transformation).

    When a problem is not solvable exactly (extracting the roots of a polynomial, solving a specific differential equation, …) the idea of perturbation theory is to obtain an approximate solution (analytically) by inserting a small parameter \epsilon and assuming the answer has the form a_0 + a_1 \epsilon + a_2 \epsilon^2 + a_3 \epsilon^3 + \dots.

    Procedure:

    • Insert \displaystyle \epsilon
    • Set \displaystyle \epsilon = 0 (the problem must be exactly solvable for \displaystyle \epsilon = 0)
    • The solution has the form \displaystyle x_{\epsilon} = a_0 + a_1 \epsilon + a_2 \epsilon^2 + a_3 \epsilon^3 + \dots
    • Set \displaystyle \epsilon = 1 (the solution of the initial problem is now \displaystyle a_0 + a_1 + a_2 + a_3 + \dots)
    • Study the convergence of \displaystyle a_0 + a_1 + a_2 + a_3 + \dots
    • If necessary use a summation method

    Solve a very simple ‘problem’ using perturbation theory:

    \displaystyle x - 2 = 0 \quad \text{Problem}
    \displaystyle x - 2\epsilon = 0 \quad \text{Insert } \epsilon
    \displaystyle (a_0 + a_1 \epsilon + a_2 \epsilon^2 + \dots) - 2\epsilon = 0 \quad \text{Insert } x_{\epsilon}
    \displaystyle a_0 + (a_1 - 2)\epsilon + a_2 \epsilon^2 + \dots = 0 + 0 + 0 + \dots

    Because coefficients of Taylor series are unique (see proof) we are allowed to compare powers of \epsilon.

    \displaystyle \implies a_0 = 0,\ a_1 = 2,\ a_2 = 0,\ \dots
    \displaystyle x_{\epsilon} = 0 + 2\epsilon + 0 + \dots
    \displaystyle x_{\epsilon} = 2\epsilon

    This implies that x = 2 (setting \epsilon = 1).

    In the figure below, we visualize the solution of x - 2\epsilon = 0 as a function of \epsilon.


    Solution of the problem x - 2 = 0 using perturbation theory. For \epsilon = 0 the solution is zero and for \epsilon = 1 the solution is 2. By making \epsilon approach 1 we obtain an increasingly precise answer to the initial problem.

  • The Path to Perturbation Theory

    “Almost no problems are exactly solvable”

    Carl Bender

    In physics, most problems—from the anharmonic oscillator to Quantum Electrodynamics (QED)— cannot be solved exactly. We rely on Perturbation Theory, expanding the solution in powers of a small coupling constant (often denoted λ or α).

    A classic example is the quantum anharmonic oscillator, where the Hamiltonian is

    \displaystyle H = \frac{p^2}{2m} + \frac{1}{2} m \omega^2 x^2 + \lambda x^4

    (with the quartic term treated as a small perturbation λ ≪ 1). Another famous case is QED, where the fine-structure constant α ≈ 1/137 serves as the small parameter for expanding processes involving electrons and photons.

    The challenge? These perturbative expansions are almost always asymptotic series with a null radius of convergence. The coefficients a_n typically grow factorially as n! (or even faster), making the series eventually diverge regardless of how small the coupling constant is.

    For instance, in the anharmonic oscillator, the perturbative corrections to the energy levels involve terms whose coefficients grow like n!, leading to a divergent series for any nonzero λ. Similarly, in QED, the perturbative series for quantities like the electron’s anomalous magnetic moment (g−2) is asymptotic: it gives spectacular agreement with experiment up to high orders, but diverges if pushed too far.

    This is where our journey pays off. By applying Borel resummation and Padé approximants to these divergent series, we can extract non-perturbative information and obtain remarkably accurate physical predictions. In quantum field theory, Borel summation helps recover meaningful results from perturbative expansions (for example, in certain Euclidean field theories or by handling contributions from instantons and renormalons). Padé approximants are routinely used to accelerate convergence and estimate the behavior of series in QCD and other models.

    In our upcoming posts, we will see how this mathematical framework becomes the essential language of modern theoretical physics.

  • Summation Techniques for Perturbation Theory

    After several months of exploring various summation and acceleration methods, it is time to synthesize these tools before we apply them to one of their most important domains: Perturbation Theory.

    When a power series \displaystyle \sum_{n=0}^\infty a_n z^{n} has a small or even zero radius of convergence, or converges too slowly to be numerically useful, we require more than standard arithmetic. Here is a recap of the methods we have covered and how they bridge the gap between formal series and numerical values.

    Summation Methods: Taming Divergence

    • Borel and Borel–Écalle: By using the Borel transform, we transform factorially growing coefficients into a convergent series in the Borel plane. The Borel–Écalle framework further allows us to handle singularities via resurgence theory, providing a unique “resummed” value even for non-Borel-summable series.
    • Euler and Zeta summation: These methods assign finite values to divergent sums by analytic continuation. While Euler summation is ideal for alternating series, Zeta summation provides a powerful way to handle the infinite sums that frequently appear in quantum vacuum energy calculations.
    • Generic summation: Beyond these classical approaches, we have also explored the idea of generic summation, which provides a unifying framework for assigning values to divergent or slowly convergent series.

    Acceleration Techniques: Enhancing Convergence

    • Shanks Transformation and Padé Approximants: These nonlinear transformations (often related to continued fractions) excel at capturing the behavior of functions beyond their radius of convergence, particularly when poles are present.
    • Richardson Extrapolation: A fundamental tool for numerical analysis that cancels out the leading error terms, allowing us to estimate the limit of a sequence Sn with much higher precision from only a few terms.

    An important consistency check underlies all these techniques: whenever different summation or acceleration methods apply to the same series, they agree on a common value. This convergence of independent approaches is not accidental, it reflects the fact that these methods capture an underlying analytic object beyond the formal series itself. When they work, they do not merely assign a value: they reveal a coherent extension of the function that the original divergent expansion was hinting at.

  • Richardson extrapolation: example

    Richardson extrapolation acts as a powerful convergence accelerator in numerical methods. Instead of relying solely on extremely fine discretizations—which can be computationally expensive—it combines results from coarser simulations to estimate the zero-step (continuum) limit.

    This idea appears in several numerical techniques (e.g., Romberg integration, mesh convergence studies in engineering), where it can significantly improve accuracy at a modest additional cost.

    When the error behaves regularly, Richardson extrapolation can dramatically reduce the error—sometimes by an order of magnitude or more—without requiring prohibitively fine meshes.

    Consider the following (slowly) converging series (see this post Zeta summation: Basel problem):

    \displaystyle 1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} + ... = \frac{\pi^2}{6}

    and calculate its corresponding Richardson extrapolation using up to 4-term partial sums:

    \displaystyle N = 2
    \displaystyle S_2 = 1 + \frac{1}{4} = \frac{5}{4}
    \displaystyle S_3 = 1 + \frac{1}{4} + \frac{1}{9} = \frac{49}{36}
    \displaystyle S_4 = 1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} = \frac{205}{144}

    The Richardson extrapolation is (see this post Richardson extrapolation):

    \displaystyle \mathcal{R}_N = \frac{N^2S_N -2(N+1)^2S_{N+1} + (N+2)^2S_{N+2}}{2}

    in this case:

    \displaystyle \mathcal{R}_2 = \frac{4 \times (\frac{5}{4}) - 2 \times 9 \times (\frac{49}{36}) + 16 \times \frac{205}{144}}{2}
    \displaystyle = \frac{5 - \frac{49}{2} + \frac{205}{9}}{2}
    \displaystyle = 1.6389

    We have:

    \displaystyle \frac{\pi^2}{6} = 1.6449
    \displaystyle \mathcal{R}_{2} = 1.6389
    \displaystyle S_4 = 1.4236

    Using solely a partial series with 4 terms, we were able to speed up the convergence of the above-mentioned series considerably using Richardson extrapolation.

  • Richardson extrapolation

    Another method for accelerating convergence is Richardson extrapolation. This method is similar to the Shanks transformation. Let’s model the partial sum as:

    \displaystyle S_N = \sum_{n=0}^{N} a_n = S + \frac{a}{N} + \frac{b}{N^2} + \frac{c}{N^3} + \dots

    where S = \lim_{N \to \infty} S_N.

    \displaystyle S_N = S + \frac{a}{N} + \frac{b}{N^2} + \frac{c}{N^3} + \dots
    \displaystyle S_{N+1} = S + \frac{a}{N+1} + \frac{b}{(N+1)^2} + \frac{c}{(N+1)^3} + \dots
    \displaystyle S_{N+2} = S + \frac{a}{N+2} + \frac{b}{(N+2)^2} + \frac{c}{(N+2)^3} + \dots

    equivalently:

    \displaystyle N^2 S_N = N^2 S + N a + b + \frac{c}{N} + \dots
    \displaystyle (N+1)^2 S_{N+1} = (N+1)^2 S + (N+1) a + b + \frac{c}{N+1} + \dots
    \displaystyle (N+2)^2 S_{N+2} = (N+2)^2 S + (N+2) a + b + \frac{c}{N+2} + \dots

    and

    \displaystyle N^2 S_N = N^2 S + N a + b + \frac{c}{N} + \dots
    \displaystyle -2 (N+1)^2 S_{N+1} = -2 (N+1)^2 S - 2 (N+1) a - 2 b - 2 \frac{c}{N+1} + \dots
    \displaystyle (N+2)^2 S_{N+2} = (N+2)^2 S + (N+2) a + b + \frac{c}{N+2} + \dots

    Summing up the system above:

    \displaystyle N^2 S_N - 2 (N+1)^2 S_{N+1} + (N+2)^2 S_{N+2} = N^2 S + N a + b + \frac{c}{N} + \dots
    \displaystyle -2 (N+1)^2 S - 2 (N+1) a - 2 b - 2 \frac{c}{N+1} + \dots
    \displaystyle + (N+2)^2 S + (N+2) a + b + \frac{c}{N+2} + \dots

    Simplifying and taking the limit N \to \infty:

    \displaystyle N^2 S_N - 2 (N+1)^2 S_{N+1} + (N+2)^2 S_{N+2} = N^2 S + a N + b
    \displaystyle - 2 N^2 S - 4 N S - 2 S - 2 a N - 2 a - 2 b
    \displaystyle + N^2 S + 4 N S + 4 S + a N + 2 a + b
    \displaystyle = 2 S
    \displaystyle \implies S = \frac{N^2 S_N - 2 (N+1)^2 S_{N+1} + (N+2)^2 S_{N+2}}{2} \quad \text{(as } N \to \infty\text{)}

    On this basis, we define the Richardson extrapolation as:

    \boxed{\displaystyle \mathcal{R}_N := \frac{N^2 S_N - 2 (N+1)^2 S_{N+1} + (N+2)^2 S_{N+2}}{2}}

  • Shanks transformation: Example 2

    To illustrate the Shanks transformation we will use the following series:

    \displaystyle \ln(x) = (x-1)-\frac{1}{2}(x-1)^2+\frac{1}{3}(x-1)^3-\dots
    \displaystyle \ln(x) = \sum_{n=1}^{\infty} (-1)^{n+1} \frac{(x-1)^n}{n}

    This series is converging very slowly. To illustrate this we will compute ln(2):

    \displaystyle \ln(2) = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} + \dots
    \displaystyle \ln(2) = \sum_{n=1}^{\infty} (-1)^{n+1} \frac{1}{n}

    The corresponding partial sum of ln(2) is:

    \displaystyle S_N = \sum_{n=1}^{N} (-1)^{n+1} \frac{1}{n}

    Let’s calculate some terms of this partial sum:

    \displaystyle S_1 = 1 \quad S_{10} = 0.6456
    \displaystyle S_2 = 0.5 \quad S_{400} = 0.6919
    \displaystyle S_3 = 0.8333 \quad \ln(2) = 0.6931

    Perform the Shanks transform (using SN = S4):

    \displaystyle S_4 = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} = \frac{12}{12} - \frac{6}{12} + \frac{4}{12} - \frac{3}{12} = \frac{7}{12}
    \displaystyle S_4^2 = \frac{49}{144}
    \displaystyle S_5 = \frac{7}{12} + \frac{1}{5} = \frac{35}{60} + \frac{12}{60} = \frac{47}{60}
    \displaystyle S_3 = 1 -\frac{1}{2} + \frac{1}{3} = \frac{6}{6} - \frac{3}{6} + \frac{2}{6} = \frac{5}{6}
    \displaystyle \mathcal{S} = \frac{S_N^2 - S_{N+1}S_{N-1}}{2S_N -S_{N+1}- S_{N-1}}
    \displaystyle \mathcal{S} = \frac{\frac{49}{144} - (\frac{47}{60} \frac{5}{6})}{\frac{14}{12} - \frac{47}{60} -\frac{5}{6}}
    \displaystyle \mathcal{S} = \frac{\frac{49}{144} - \frac{235}{360}}{\frac{70}{60} - \frac{47}{60} -\frac{50}{60}}
    \displaystyle \mathcal{S} = \frac{-225/720}{-27/60}
    \displaystyle \mathcal{S} = 0.6944

    Since S_{358} = 0.6918 and \ln(2) = 0.6931 It is necessary to calculate up to 358 terms of the partial sum SN to obtain a relative error equal to that of the Shanks transformation \mathcal{S} based on the first 4 terms.

  • Shanks transformation: Example 1

    Let’s transform the geometric series:

    \displaystyle 1+x+x^2+x^3+x^4+\dots

    using the corresponding partial sum S2:

    \displaystyle 1+x+x^2
    \displaystyle S_2^2 = (1+x+x^2)^2 = 1 + 2x + 3x^2 + 2x^3 + x^4
    \displaystyle S_3 S_1 = (1+x+x^2+x^3)(1+x) = 1 + 2x + 2x^2 + 2x^3 + x^4
    \displaystyle 2S_2 = 2 + 2x + 2x^2
    \displaystyle S_3 = 1+x+x^2+x^3
    \displaystyle S_1 = 1 + x
    \displaystyle \mathcal{S} = \frac{(1+2x+3x^2+2x^3+x^4) - (1+2x+2x^2+2x^3+x^4)}{(2+2x +2x^2) - (1+x+x^2+x^3) - (1+x)}
    \displaystyle \mathcal{S} = \frac{x^2}{x^2-x^3}
    \displaystyle \mathcal{S} = \frac{1}{1-x}

    Therefore the Shanks transformation of the geometric series is correctly \frac{1}{1-x}. A similar result can be obtained for the series:

    \displaystyle 1-x+x^2-x^3+\dots

    The corresponding Shanks transformation is \frac{1}{1+x}.

    Remarkably, the Shanks transformation yields the exact sum of the geometric series using only a few terms, since the error model S_N = S + \alpha \beta^N perfectly describes such a series. The Gregory series for \pi is notoriously slow, requiring five hundred terms just to calculate two correct decimal places. However, by applying the Shanks transformation, we can accelerate this convergence dramatically, extracting high-precision results from only a few initial partial sums. This demonstration highlights how a simple mathematical shift can transform an inefficient infinite series into a powerful computational tool.

  • Accelerate convergence (Shanks Transformation)

    Imagine you’ve solved a problem using a Taylor series expansion of the solution, and the resulting series converges very slowly. The Shanks transformation is one way of improving the speed of convergence of a convergent series.
    Given the partial sum:

    \displaystyle S_N = a_0 + a_1 + a_2 + \dots + a_N

    Suppose that this partial sum converges to:

    \displaystyle S = a_0 + a_1 + a_2 + \dots
    \displaystyle S = \lim_{N \to \infty} \sum_{k=0}^{N} a_k

    We can model the difference between S and SN as follows:

    \displaystyle S_N = S + \alpha \beta^{N}

    Where |\beta| < 1:

    \displaystyle S_{N+1} = S + \alpha \beta^{N+1}
    \displaystyle S_N = S + \alpha \beta^{N}
    \displaystyle S_{N-1} = S + \alpha \beta^{N-1}

    and

    \displaystyle S_{N+1} - S = \alpha \beta^{N+1}
    \displaystyle S_N - S = \alpha \beta^{N}
    \displaystyle S_{N-1} - S = \alpha \beta^{N-1}

    Therefore:

    \displaystyle \frac{S_N - S}{S_{N-1}-S} = \frac{\alpha \beta^{N}}{\alpha \beta^{N-1}} = \beta
    \displaystyle \frac{S_{N+1} - S}{S_{N}-S} = \frac{\alpha \beta^{N+1}}{\alpha \beta^{N}} = \beta
    \displaystyle \frac{S_N - S}{S_{N-1}-S} = \frac{S_{N+1} - S}{S_{N}-S}
    \displaystyle (S_N - S)(S_{N}-S) = (S_{N+1} - S)(S_{N-1}-S)
    \displaystyle S_N^2 - 2S_NS + S^2 = S_{N+1}S_{N-1}- SS_{N+1} - SS_{N-1} + S^2
    \displaystyle S_N^2 - 2S_NS + S^2= S_{N+1}S_{N-1} + S^{2}- S(S_{N+1}+ S_{N-1})
    \displaystyle S_N^2 - 2S_NS = S_{N+1}S_{N-1}- S(S_{N+1}+ S_{N-1})
    \displaystyle S_N^2 - S_{N+1}S_{N-1} = - S(S_{N+1}+ S_{N-1}) - 2S_NS
    \displaystyle S_N^2 - S_{N+1}S_{N-1} = S(-S_{N+1}- S_{N-1})+ 2S_N
    \displaystyle S = \frac{S_N^2 - S_{N+1}S_{N-1}}{2S_N -S_{N+1}- S_{N-1}}

    According to this result, the Shanks transformation is defined as:

    \displaystyle \mathcal{S} := \frac{S_N^2 - S_{N+1}S_{N-1}}{2S_N -S_{N+1}- S_{N-1}}

  • Summation summary

    Series Standard Euler Borel Borel–Écalle Zeta
    \displaystyle 1+\frac{1}{4}+\frac{1}{9}+\dots
    \displaystyle \frac{\pi^2}{6}
    \displaystyle \frac{\pi^2}{6}
    \displaystyle \frac{\pi^2}{6}
    \displaystyle \frac{\pi^2}{6}
    \displaystyle \frac{\pi^2}{6}
    \displaystyle 1-1+1-1+\dots
    \displaystyle \frac12
    \displaystyle \frac12
    \displaystyle \frac12
    \displaystyle \frac12
    \displaystyle 1!-2!+3!-4!+\dots
    \displaystyle \approx 0.5963
    \displaystyle \approx 0.5963
    \displaystyle 1+2+3+4+\dots
    \displaystyle -\frac{1}{12}
    \displaystyle 1!+2!+3!+4!+\dots

    The various summation methods presented in this table are not arbitrary assignments of values to divergent series; they are governed by the fundamental principle of regularity and consistency. A method is “regular” if it preserves the sum of any convergent series, and “consistent” if different methods yield the same value when they both apply to a given series.

    In theoretical physics and asymptotic analysis, these techniques are indispensable. Borel and Euler summations are frequently used to handle “weakly” divergent perturbative expansions in quantum mechanics. Zeta-function regularization is a cornerstone of modern physics, particularly in the calculation of the Casimir effect and in string theory, where it provides a rigorous way to subtract infinities from physical observables. Meanwhile, the Borel–Écalle theory of resurgent functions offers the most sophisticated framework for “decoding” the non-perturbative information hidden within divergent power series. By bridging the gap between divergent expansions and their underlying analytic structures, these methods allow mathematicians and physicists to extract precise physical predictions from seemingly ill-defined mathematical objects.